> higher inflation implying that “rates should go up” is central bank doctrine

Uh, no. If you have no central bank, more consumption and more employment means more demand for money. Ceteris paribus, that will raise rates. (Our own history with free banking is more complicated since the only inflationary period was driven by specie introduction from California's gold rush. The predominant problem in antebellum America was deflation and bank collapses.)

You're correct inasmuch as central banks quicken this reaction, and–when done properly–dampen it. But the fundamental engine is emergent, at least for nominal rates.